show that the linear mapping C ∞ (X, G) ’ L(»X, G) is bounded we can reformu-

late this equivalently using (3.12), the universal property of »X and the uniform

boundedness principle (5.18) in turn:

C ∞ (X, G) ’ L(»X, G) is L

⇐’ »X ’ L(C ∞ (X, G), G) is L

⇐’ X ’ L(C ∞ (X, G), G) is C ∞

evf

⇐’ X ’ L(C ∞ (X, G), G) ’ G is C ∞

and since the composition is just f we are done.

Conversely we have to show that L(»X, G) ’ C ∞ (X, G) belongs to L. Composed

with evx : C ∞ (X, G) ’ G this yields the bounded linear map evδ(x) : L(»X, G) ’

G. Thus this follows from the uniform boundedness principle (5.26).

That δX is initial follows immediately from the fact that the structure of X is initial

with respect to family {f = evo —¦δX : f ∈ C ∞ (X, R)}.

Remark. The corresponding result with the analogous proof is true for holomor-

phic Fr¨licher spaces, Lipk -spaces, and ∞ -spaces. For the ¬rst see [Siegl, 1997] for

o

the last two see [Fr¨licher, Kriegl, 1988].

o

23.7. Corollary. Let X be a Fr¨licher space such that the functions in C ∞ (X, R)

o

separate points on X. Then X is di¬eomorphic as Fr¨licher space to a subspace

o

of the convenient vector space »(X) ⊆ C ∞ (X, R) with the initial smooth structure

(generated by the restrictions of linear bounded functionals, among other possibili-

ties).

We have constructed the free convenient vector space »X as the c∞ -closure of the

linear subspace generated by the point evaluations in C ∞ (X, R) . This is not very

constructive, in particular since adding Mackey-limits of sequences (or even nets)

of a subspace does not always give its Mackey-closure. In important cases (like

when X is a ¬nite dimensional smooth manifold) one can show however that not

only »X = C ∞ (X, R) , but even that every element of »X is the Mackey-limit of

a sequence of linear combinations of point evaluations, and that C ∞ (X, R) is the

space of distributions of compact support.

23.8. Proposition. Let E be a convenient vector space and X a ¬nite dimensional

smooth separable manifold. Then for every ∈ C ∞ (X, E) there exists a compact

set K ⊆ X such that (f ) = 0 for all f ∈ C ∞ (X, E) with f |K = 0.

Proof. Since X is separable its compact bornology has a countable basis {Kn :

n ∈ N} of compact sets. Assume now that no compact set has the claimed property.

Then for every n ∈ N there has to exist a function fn ∈ C ∞ (X, E) with fn |Kn = 0

23.8

23.10 23. Fr¨licher spaces and free convenient vector spaces

o 243

n

but (fn ) = 0. By multiplying fn with (fn ) we may assume that (fn ) = n. Since

every compact subset of X is contained in some Kn one has that {fn : n ∈ N} is

bounded in C ∞ (X, E), but ({fn : n ∈ N} is not; this contradicts the assumption

that is bounded.

23.9. Remark. The proposition above remains true if X is a ¬nite dimensional

smooth paracompact manifold with non-measurably many components. In order

to show this generalization one uses that for the partition {Xj : j ∈ J} by the

non-measurably many components one has C ∞ (X, E) ∼ j∈J C ∞ (Xj , E), and the

=

fact that an belongs to the dual of such a product if it is a ¬nite sum of elements

of the duals of the factors. Now the result follows from (23.8) since the components

of a paracompact manifold are paracompact and hence separable.

For such manifolds X the dual C ∞ (X, R) is the space of distributions with compact

support. In fact, in case X is connected, C ∞ (X, R) is the space of all linear func-

tionals which are continuous for the classically considered topology on C ∞ (X, R)

by (6.1); and in case of an arbitrary X this result follows using the isomorphism

C ∞ (X, R) ∼ j C ∞ (Xj , R) where the Xj denote the connected components of X.

=

23.10. Theorem. [Fr¨licher, Kriegl, 1988], 5.1.7 Let E be a convenient vector

o

space and X a ¬nite dimensional separable smooth manifold. Then the Mackey-

adherence of the linear subspace generated by { —¦evx : x ∈ X, ∈ E } is C ∞ (X, E) .

Proof. The proof is in several steps.

(Step 1) There exist gn ∈ C ∞ (R, R) with supp(gn ) ⊆ [’ n , n ] such that for every

22

f ∈ C ∞ (R, E) the set {n· f ’ k∈Z f (rn,k )gn,k : n ∈ N} is bounded in C ∞ (R, E),

where rn,k := 2k and gn,k (t) := gn (t ’ rn,k ).

n

We choose a smooth h : R ’ [0, 1] with supp(h) ⊆ [’1, 1] and k∈Z h(t ’ k) = 1

for all t ∈ R and we de¬ne Qn : C ∞ (R, E) ’ C ∞ (R, E) by setting

Qn (f )(t) := k

f ( n )h(tn ’ k).

k

Let K ⊆ R be compact. Then

n(Qn (f ) ’ f )(t) = k 1

(f ( n ) ’ f (t)) · n · h(tn ’ k) ∈ B1 (f, K + supp(h))

n

k

for t ∈ K, where Bn (f, K1 ) denotes the absolutely convex hull of the bounded set

n

δ n f (K1 ).

To get similar estimates for the derivatives we use convolution. Let h1 : R ’ R be

a smooth function with support in [’1, 1] and R h1 (s)ds = 1. Then for t ∈ K one

has

(f — h1 )(t) := f (t ’ s)h1 (s)ds ∈ B0 (f, K + supp(h1 )) · h1 1 ,

R

where h1 1 := R |h1 (s)|ds. For smooth functions f, h : R ’ R one has (f —h)(k) =

f —h(k) ; one immediately deduces that the same holds for smooth functions f : R ’

23.10

244 Chapter V. Extensions and liftings of mappings 23.10

E and one obtains (f — h1 )(t) ’ f (t) = R (f (t ’ s) ’ f (t))h1 (s)ds ∈ diam(supp(h1 )) ·

h1 1 · B1 (f, K + supp(h1 )) for t ∈ K, where diam(S) := sup{|s| : s ∈ S}. Using

now hn (t) := n · h1 (nt) we obtain for t ∈ K:

(Qm (f ) — hn ’ f )(k) (t) = (Qm (f ) — h(k) ’ f — h(k) )(t) + (f (k) — hn ’ f (k) )(t)

n n

= (Qm (f ) ’ f ) — h(k) (t) + (f (k) — hn ’ f (k) )(t)

n

∈ B0 (Qm (f ) ’ f, K + supp(hn )) · h(k) 1 +

n

+ B1 (f (k) , K + supp(hn )) · diam(supp(hn )) · hn 1

(k)

1k 1

⊆ · B1 (f, K + supp(hn ) + supp(h)) · h1

mn 1

m

+ n · B1 (f (k) , K + supp(hn )) · hn 1 .

Let now m := 2n and P n (f ) := Qm (f ) — hn . Then

1

(k) (k)

(t) ∈ nk+1 2’n · B1 (f, K + ( n +

n · P n (f ) ’ f 1

)[’1, 1]) h1 1

2n

+ B1 (f (k) , K + n [’1, 1]) h1 1

1

for t ∈ K and the right hand side is uniformly bounded for n ∈ N.

h(s2n ’ k)hn (t + k2’n ’ s)ds = h(s2n )hn (t ’ s)ds we obtain

With gn (t) := R R

n

f (k2’n )h(t2n ’ k) — hn

P n (f )(t) = (Q2 (f ) — hn )(t) =

k

f (k2’n ) h(s2n ’ k)hn (t ’ s)ds

=

R

k

f (k2’n )gn (t ’ k2’n ).

=

k

Thus rn,k := k2’n and the gn have all the claimed properties.

(Step 2) For every m ∈ N and every f ∈ C ∞ (Rm , E) the set

n· f ’ f (rn;k1 ,...,km )gn;k1 ,...,km : n ∈ N

k1 ∈Z,...,km ∈Z

is bounded in C ∞ (Rm , E), where rn;k1 ,...,km := (rn,k1 , . . . , rn,km ) and

gn;k1 ,...,km (x1 , . . . , xm ) := gn,k1 (x1 ) · . . . · gn,km (xm ).

We prove this statement by induction on m. For m = 1 it was shown in step 1.

Now assume that it holds for m and C ∞ (R, E) instead of E. Then by induction

hypothesis applied to f ∨ : C ∞ (Rm , C ∞ (R, E)) we conclude that

n· f ’ )gn;k1 ,...,km : n ∈ N

f (rn;k1 ,...,km ,

k1 ∈Z,...,km ∈Z

23.10

23.12 23. Fr¨licher spaces and free convenient vector spaces

o 245

is bounded in C ∞ (Rm+1 , E). Thus it remains to show that

f (rn;k1 ,...,km , rkm+1 )gn,km+1 : n ∈ N

n gn;k1 ,...,km f (rn;k1 ,...,km , )’

k1 ,...,km km+1

is bounded in C ∞ (Rm+1 , E). Since the support of the gn;k1 ,...,km is locally ¬nite

only ¬nitely many summands of the outer sum are non-zero on a given compact set.

Thus it is enough to consider each summand separately. By step (1) we know that

the linear operators h ’ n h ’ k h(rn,k )gn,k , n ∈ N, are pointwise bounded.

So they are bounded on bounded sets, by the linear uniform boundedness principle

(5.18). Hence

n · f (rn;k1 ,...,km , )’ f (rn;k1 ,...,km , rkm+1 )gn,km+1 : n ∈ N

km+1

is bounded in C ∞ (Rm+1 , E). Using that the multiplication R — E ’ E is bounded

one concludes immediately that also the multiplication with a map g ∈ C ∞ (X, R)

is bounded from C ∞ (X, E) ’ C ∞ (X, E) for any Fr¨licher space X. Thus the proof

o

of step (2) is complete.

(Step 3) For every ∈ C ∞ (X, E) there exist xn,k ∈ X and n,k ∈ E such that

{n( ’ k n,k —¦ evxn,k ) : n ∈ N} is bounded in C ∞ (X, E) , where in the sum only

¬nitely many terms are non-zero. In particular the subspace generated by E —¦ evx

for E ∈ E and x ∈ X is c∞ -dense.

By (23.8) there exists a compact set K with f |K = 0 implying (f ) = 0. One

can cover K by ¬nitely many relatively compact Uj ∼ Rm (j = 1 . . . N ). Let

=

{hj : j = 0 . . . N } be a partition of unity subordinated to {X K, U1 , . . . , UN }.

N

Then (f ) = j=1 (hj · f ) for every f . By step (2) the set

n(hj f ’ hj f (rn,k1 ,...,km )gn,k1 ,...,km : n ∈ N

is bounded in C ∞ (Uj , E). Since supp(hj ) is compact in Uj this is even bounded in

C ∞ (X, E) and for ¬xed n only ¬nitely many rn,k1 ,...,km belong to supp(hj ). Thus

the above sum is actually ¬nite and the supports of all functions in the bounded

subset of C ∞ (Uj , E) are included in a common compact subset. Applying to this

subset yields that n ( (hj f ) ’ n,k1 ,...,km —¦ ev(rn,k1 ,...,km ) : n ∈ N is bounded

in R, where n,k1 ,...,km (x) := hj (rn,k1 ,...,km )gn;k1 ,...,km · x .

To complete the proof one only has to take as xn,k all the rn,k1 ,...,km for the ¬nitely

many charts Uj ∼ Rm and as n,k the corresponding functionals n,k1 ,...,km ∈ E .

=

23.11. Corollary. [Fr¨licher, Kriegl, 1988], 5.1.8 Let X be a ¬nite dimensional

o

separable smooth manifold. Then the free convenient vector space »X over X is

equal to C ∞ (X, R) .

23.12. Remark. In [Kriegl, Nel, 1990] it was shown that the free convenient

vector space over the long line L is not C ∞ (L, R) and the same for the space E of

points with countable support in an uncountable product of R.

23.12

246 Chapter V. Extensions and liftings of mappings 23.13

In [Adam, 1995, 2.2.6] it is shown that the isomorphism δ — : L(C ∞ (X, R) , G) ∼ =

C ∞ (X, G) is even a topological isomorphism for (the) natural topologies on all

spaces under consideration provided X is a ¬nite dimensional separable smooth

manifold. Furthermore, the corresponding statement holds for holomorphic map-

pings, provided X is a separable complex manifold modeled on polycylinders. For

Riemannian surfaces X it is shown in [Siegl, 1997, 2.11] that the free convenient

vector space for holomorphic mappings is the Mackey adherence of the linear sub-

space of H(X, C) generated by the point evaluations evx for x ∈ X. In [Siegl, 1997,

2.52] the same is shown for pseudo-convex subsets of X ⊆ Cn . Re¬‚exivity of the

space of scalar valued functions implies that the linear space generated by the point

evaluations is dense in the dual of the function space with respect to its bornolo-

gical topology by [Siegl, 1997, 3.3]. And conversely if Λ(X) is this dual, then the

function space is re¬‚exive. Thus Λ(E) = C ∞ (E, R) for non-re¬‚exive convenient

vector spaces E. Partial positive results for in¬nite dimensional spaces have been

obtained in [Siegl, 1997, section 3].

23.13. Remark. On can de¬ne convenient co-algebras dually to convenient alge-

bras, as a convenient vector space E together with a compatible co-algebra struc-

ture, i.e. two bounded linear mappings

µ : E ’ E —β E, called co-multiplication, into the c∞ -completion (4.29) of

˜

the bornological tensor product (5.9);

and µ : E ’ R, called co-unit,

such that one has the following commutative diagrams:

w w E—˜ (E—˜ E)

˜ ∼

µ—β Id =

˜ ˜ ˜

E —β E (E —β E)—β E β β

u u

˜

µ Id —β µ

w E—˜ E

µ

E β

ee

˜

E —β E

j

h

h e˜ Id

g

e

h µ—

µ

h

w R—˜ E

∼

=

E β

In words, the co-multiplication has to be co-associative and µ has to be a co-unit

with respect to µ.

If, in addition, the following diagram commutes

w E—˜ E

∼

U

R

=

RR

˜

E —β E

j

h

β

h

R hh

µ µ

E

then the co-algebra is called co-commutative.

23.13

24.1 24. Smooth mappings on non-open domains 247

Morphisms g : E ’ F between convenient co-algebras E and F are bounded linear

mappings for which the following diagrams commute:

w F —˜u F w Ru

u

˜

g —g

u

Id

˜

E —β E R

β

µE µF

µE µF

wF wF

g g

E E

A co-idempotent in a convenient co-algebra E, is an element x ∈ E satisfying

µ(x) = 1 and µ(x) = x — x. They correspond bijectively to convenient co-algebra

morphisms R ’ E, see [Fr¨licher, Kriegl, 1988, 5.2.7].

o

In [Fr¨licher, Kriegl, 1988, 5.2.4] it was shown that »(X — Y ) ∼ »(X)—»(Y ) using

˜

o =

only the universal property of the free convenient vector space. Thus »(∆) : »(X) ’

»(X — X) ∼ »(X)—»(X) of the diagonal mapping ∆ : X ’ X — X de¬nes a co-

˜

=

multiplication on »(X) with co-unit »(const) : »(X) ’ »({—}) ∼ R. In this way »

=

becomes a functor from the category of Fr¨licher spaces into that of convenient co-

o

algebras, see [Fr¨licher, Kriegl, 1988, 5.2.5]. In fact this functor is left-adjoint to the

o

functor I, which associates to each convenient co-algebra the Fr¨licher space of co-

o

idempotents with the initial structure inherited from the co-algebra, see [Fr¨licher,

o

Kriegl, 1988, 5.2.9].

Furthermore, it was shown in [Fr¨licher, Kriegl, 1988, 5.2.18] that any co-idempo-

o

eve

tent element e of »(X) de¬nes an algebra-homomorphism C ∞ (X, R) ∼ »(X) ’ ’ ’

=

R. Thus the equality I(»(X)) = X, i.e. every co-idempotent e ∈ »(X) is given by

evx for some x ∈ X, is thus satis¬ed for smoothly realcompact spaces X, as they

are treated in chapter IV.

24. Smooth Mappings on Non-Open Domains

In this section we will discuss smooth maps f : E ⊇ X ’ F , where E and F

are convenient vector spaces and X are certain not necessarily open subsets of E.

We consider arbitrary subsets X ⊆ E as Fr¨licher spaces with the initial smooth

o

structure induced by the inclusion into E, i.e., a map f : E ⊇ X ’ F is smooth if

and only if for all smooth curves c : R ’ X ⊆ E the composite f —¦ c : R ’ F is a

smooth curve.

24.1. Lemma. Convex sets with non-void interior.

Let K ⊆ E be a convex set with non-void c∞ -interior K o . Then the segment

(x, y] := {x + t(y ’ x) : 0 < t ¤ 1} is contained in K o for every x ∈ K and y ∈ K 0 .

The interior K o is convex and open even in the locally convex topology. And K is

closed if and only if it is c∞ -closed.

Proof. Let y0 := x + t0 (y ’ x) be an arbitrary point on the segment (x, y], i.e.,

0 < t0 ¤ 1. Then x+t0 (K o ’x) is an c∞ -open neighborhood of y0 , since homotheties

are c∞ -continuous. It is contained in K, since K is convex.

24.1